# Alternate current and complex description

• Lindsayyyy
In summary: An alternate current has the first temporal dependence ( sin(omega t)) and is always a real number.The second one is a complex number which is used as an auxiliary variable that allows solving the problem "easily".Thus, we shouldn't label both of them as I(t) because the two expressions are different.In summary, alternate current is a type of current that has a different temporal dependence (sin(omega t)) than regular current. It can be written with the sin(omega t) equation or with the euler's formula. The imaginary part of the complex number tells us that it is a constant, physical value. It is time dependent, just like real current is
Lindsayyyy
Hi everyone I hope I'm in the right board for this:

When it comes to alternate current, a complex description is pretty common as it's apperently easier to calculate with. But I don't understand alternate current to be honest. I have a problem with the basics, especially the following:

I can describe alternate current with sine, for example:

$$I(t)=I_{0}*sin(\omega t)$$

where I_0 is the max amplitude.

but I can also write an alternate current like this:
$$I(t)=I_{0}*e^{i \omega t}$$
i=imaginary unit
but with Euler's formula I can reqrite this to:

$$I(t)=I_{0}*[cos(\omega t)+i*sin(\omega t)]$$

What does the imaginary part tells me here? Both equations express an alternate current, but they are different (to me at least:) )

I'm pretty sure I have a wrong imagination of this stuff. Can someone explain it to me? I hope you get the point where my problems are.

An alternate current has the first temporal dependence ( sin(omega t)) and is always a real number.
The second one is a complex number which is used as an auxiliary variable that allows solving the problem "easily".
Thus, we shouldn't label both of them as I(t) because the two expressions are different.

Hi Lindsayyyy!

Current and complex current are two completely different things

Current is variable (it depends on t).

Complex current is constant.

From the PF Library on https://www.physicsforums.com/library.php?do=view_item&itemid=303"

In a steady sinusoidal (AC) circuit of frequency ω, the (instantaneous) voltage and current $V\text{ and }I$ can always be written:
$V =\ V_x\cos\omega t + V_y\sin\omega t$ and $I =\ I_x\cos\omega t + I_y\sin\omega t$​

Then the complex voltage and complex current between any two points are the constants defined as $\mathbf{V} =\ V_x+jV_y\text{ and }\mathbf{I} =\ I_x+jI_y$.​

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Hey tiny-tim, you've got me confused saying complex current is a constant.
I thought complex current is not so much constant, but its absolute value is constant.

Complex current is still time dependent, just like real current is.
More precisely real current is the "real" part of the complex current.To the OP, consider for instance a pendulum.

Its displacement is a cosine of the time.
When the bob is in the position of maximum displacement, all its energy is potential energy.
When the displacement is zero, all the energy of the bob is kinetic.
These two alternate with each other, but at every time the total energy is constant.

So we can describe the motion of the pendulum with a complex number.
The real part would be the actual displacement, related to potential energy.
While the imaginary part would indicate the speed of the bob, related to kinetic energy.

If we multiply the complex value with the appropriate constant, the squared absolute value is the total energy.

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Thanks for the help thus far. My problem is I don't understand how exactly I should use the complex version and especially for what I need it. The complex one isn't a physical value because we learned physical values are alway real :/ I'm confused

Lindsayyyy said:
Thanks for the help thus far. My problem is I don't understand how exactly I should use the complex version and especially for what I need it. The complex one isn't a physical value because we learned physical values are alway real :/ I'm confused

Yes, physical values are always real.
So if you need the physical value from a complex one, you take the "real" part of the complex number (the part that does not have i in it).
(Do you know what I mean with the "real" part?)

You would use the complex version just like the real version.
Meaning you can multiply, add and subtract them.

As for when you need it. It's basically an arbitrary choice.
Both forms work.
But some calculations are less work with complex values.

I like Serena said:
Yes, physical values are always real.
So if you need the physical value from a complex one, you take the "real" part of the complex number (the part that does not have i in it).
(Do you know what I mean with the "real" part?)

You would use the complex version just like the real version.
Meaning you can multiply, add and subtract them.

As for when you need it. It's basically an arbitrary choice.
Both forms work.
But some calculations are less work with complex values.

Yes, I know what you mean by real part. Did I understand it right you say:
If I have an alternate current given as a complexe number. I just take the real part and ignore the imaginary one?

Lindsayyyy said:
Yes, I know what you mean by real part. Did I understand it right you say:
If I have an alternate current given as a complexe number. I just take the real part and ignore the imaginary one?

Hmm, it's a little more complex (no pun intended ).

If you would measure the current with an ammeter, you'll measure the real part, so in this case the imaginary part is "ignored".

If you calculate the effect of a capacitor, its impedance is typically given as a complex number, depending on the frequency of the current ($Z = \frac 1 {i \omega C}$).
So if you calculate the voltage from the current through the capacitor with the complex form of Ohm's law (V = I Z), you need to calculate this with complex numbers.
If you "ignore" the imaginary part of the current in this calculation, you'll get the wrong "real" voltage in your answer.

ok, I think I understand it a bit better now, thank you.

one more question:

We can write the impendance as:

$$Z=R+iX$$

i:imaginary unit

Now what's the R exactly? Does a capacitor or indctor exist of an ohm resistance R and reactence X or is Z just the sum of the components in an electric circuit?

Capacitors are usually described as "ideal" lossless components, but when losses are taken into account it is usually via the so-called ESR parameter (equivalent series resistance). However, this is a "mode", the R does not correspond to anything specific but is just used to model all the non-ideal behaviour of a real capacitor.

However, if you write down the impedance of an inductor it is easy to see that the imaginary part of the impedance is due to the inductance, and the real part (R) is due to the resistance of the wire (which can be significant in large coils).

ok, thanks everyone for the help and patience. I have to do some exercises on this when I'm through with electrostatics.Be prepared to get annoyed

We'll be waiting with anticipation!

## 1. What is alternate current (AC)?

Alternate current is an electrical current that periodically reverses direction, unlike direct current (DC) which flows in only one direction. In AC, the flow of electricity changes direction at a certain frequency, typically 50 or 60 times per second.

## 2. How is AC described using complex numbers?

AC can be described using complex numbers because it varies sinusoidally with time. This means that the amplitude and direction of the current at any given time can be represented by a complex number on the complex plane.

## 3. What are the advantages of using AC over DC?

AC has several advantages over DC, including the ability to be transformed to different voltage levels using transformers, the ability to be transmitted over long distances with minimal energy loss, and the ability to power multiple devices simultaneously through a single power source.

## 4. How is AC used in everyday life?

AC is used in everyday life for a variety of purposes, such as powering household appliances, lighting our homes and buildings, and running industrial machinery. It is also used in the generation and transmission of electricity to power entire cities and countries.

## 5. Can AC and DC be converted to one another?

Yes, AC and DC can be converted to one another using devices called rectifiers and inverters. Rectifiers convert AC to DC, while inverters convert DC to AC. These devices are commonly used in electronic devices and power systems to convert between the two types of currents.

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